Linear-Based Eulerian Motion Modulation

ABSTRACT

In one embodiment, a method of amplifying temporal variation in at least two images comprises examining pixel values of the at least two images. The temporal variation of the pixel values between the at least two images can be below a particular threshold. The method can further include applying signal processing to the pixel values.

GOVERNMENT SUPPORT

This invention was made with government support under 1111415 from theNational Science Foundation. The government has certain rights in theinvention.

BACKGROUND OF THE INVENTION

The human visual system has limited spatio-temporal sensitivity, butmany signals that fall below its sensitivity threshold can beinformative. For example, human skin color varies slightly correspondingto blood circulation. This variation, while invisible to the naked eye,can be exploited to extract pulse rate or to visualize blood flow.Similarly, motion with low spatial amplitude, while hard or impossiblefor humans to see, can be magnified to reveal interesting mechanicalbehavior.

SUMMARY OF THE INVENTION

In one embodiment, a method of amplifying temporal variation in at leasttwo images comprises examining pixel values of the at least two images.The temporal variation of the pixel values between the at least twoimages can be below a particular threshold. The method can furtherinclude applying signal processing to the pixel values.

In another embodiment, applying signal processing can amplify thevariations of the pixel values between the at least two images. Thesignal processing can be temporal processing. The temporal processingcan be a bandpass filter or be performed by a bandpass filter. Thebandpass filter can be configured to analyze frequencies over time.Applying signal processing can include spatial processing. Spatialprocessing can remove noise.

In another embodiment, the method can include visualizing a pattern offlow of blood in a body shown in the at least two images.

In another embodiment, a system for amplifying temporal variation in atleast two images can include a pixel examination module configured toexamine pixel values of the at least two images. The temporal variationof the pixel values between the at least two images can be below aparticular threshold. The system can further include a signal processingmodule configured to apply signal processing to the pixel values.

The signal processing module can be further configured to amplifyvariations of the pixel values between the at least two images. Thesignal processing module can be further configured to employ temporalprocessing. The temporal processing can be a bandpass filter orperformed by a bandpass filter. The bandpass filter can be configured toanalyze frequencies over time. The signal processing module can befurther configured to perform spatial processing. The spatial processingcan remove noise.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawings will be provided by the Office upon request and paymentof the necessary fee.

The foregoing will be apparent from the following more particulardescription of example embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingembodiments of the present invention.

FIGS. 1A-H are example frames illustrating Eulerian Video Magnificationframework for visualizing the human pulse in a series of frames.

FIGS. 1I-J illustrate how Eulerian Video Magnification amplifies theperiodic color variation.

FIG. 2 is an illustration of a temporal processing module employing theEulerian video magnification framework.

FIG. 3 is a chart illustrating that temporal filtering can approximatespatial translation.

FIGS. 4A-D are charts illustrating motion amplification on a 1D signalfor different spatial frequencies and a values.

FIGS. 5A-B are charts illustrating motion magnification error, computedas the L₁-norm between the true motion-amplified signal (FIG. 4A) andthe temporally-filtered result (FIG. 4B), as function of wavelength, fordifferent values of δ(t), (FIG. 5A), and α, (FIG. 5B).

FIG. 6 is a graph illustrating amplification factor, α, as a function ofspatial wavelength λ, for amplifying motion.

FIGS. 7A-B are illustrations showing Eulerian video magnification usedto amplify subtle motions of blood vessels arising from blood flow.

FIG. 8 illustrates representative frames 810, 820, 830, 840, 850, and860 from additional videos demonstrating the process.

FIGS. 9A-D are graphs illustrating the frequency responses of some ofthe temporal filters.

FIGS. 10A-B are diagrams illustrating selective motion amplification ona synthetic sequence.

FIG. 11A illustrates frame from the face video (FIG. 1) with whiteGaussian noise added.

FIG. 11B illustrates intensity traces over time for the pixel markedblue on the input frame, showing the trace obtained when the (noisy)sequence is processed with the same spatial filter used to process theoriginal face sequence, a separable binomial filter of size 20.

FIG. 11C shows the trace when using a filter tuned according to theestimated radius in Eq. 15, a binomial filter of size 80.

FIGS. 12A-C are diagrams illustrating a comparison between Eulerian andLagrangian motion magnification on a synthetic sequence with additivenoise.

FIG. 13 is a flow diagram illustrating an example embodiment of theEulerian embodiment of the method described herein.

FIG. 14 illustrates a computer network or similar digital processingenvironment in which the present invention may be implemented.

FIG. 15 is a diagram of the internal structure of a computer (e.g.,client processor/device or server computers) in the computer system ofFIG. 14.

DETAILED DESCRIPTION OF THE INVENTION

A description of example embodiments of the invention follows.

The method described herein (a) reveals temporal variations in videosthat are difficult or impossible to see with the naked eye and (b)displays the temporal variations in an indicative manner. The method,called Eulerian Video Magnification, inputs a standard video sequence,applies spatial decomposition, and applies temporal filtering to theframes. The resulting signal is then amplified to reveal hiddeninformation. This method can, for example, visualize flow of bloodfilling a face in a video and also amplify and reveal small motions. Themethod can run in real time to show phenomena occurring at temporalfrequencies selected by the user.

FIGS. 1A-H are example frames illustrating Eulerian Video Magnificationframework for visualizing the human pulse in a series of frames. FIGS.1A-D illustrate four frames from the original video sequence showing aface. FIGS. 1E-H illustrate four frames with the subject's pulse signalamplified. FIG. 1I is a chart illustrating a vertical scan line from theinput and FIG. 1J is a chart illustrating a vertical scan line from theoutput videos plotted over time. FIGS. 1I-J illustrate how EulerianVideo Magnification amplifies the periodic color variation shown inspatiotemporal YT slices. In the input sequence (FIGS. 1A-D) the signalis imperceptible, but the variation is clear in the magnified sequence.

The success of these tools motivates the development of new techniquesto reveal invisible signals in videos. A combination of spatial andtemporal processing of videos can amplify subtle variations that revealimportant aspects of the world.

The method described herein considers a time series of color values atany spatial location (e.g., a pixel) and amplifies variation in a giventemporal frequency band of interest. For example, in FIG. 1, the methodselects and then amplifies a band of temporal frequencies includingplausible human heart rates. The amplification reveals the variation ofredness as blood flows through the face. For this application, lowerspatial frequencies are temporally filtered (spatial pooling) to allow asubtle input signal to rise above the camera sensor and quantizationnoise.

The temporal filtering approach not only amplifies color variation, butcan also reveal low-amplitude motion. For example, the method canenhance the subtle motions around the chest of a breathing baby. Themethod's mathematical analysis employs a linear approximation related tothe brightness constancy assumption used in optical flow formulations.The method also derives the conditions under which this approximationholds. This leads to a multiscale approach to magnify motion withoutfeature tracking or motion estimation.

Previous attempts have been made to unveil imperceptible motions invideos. For example, a method, described in Liu et al. 2005, analyzesand amplifies subtle motions and visualizes deformations that areotherwise invisible. Another method, described in Wang et al. 2006,proposes using a Cartoon Animation Filter to create perceptuallyappealing motion exaggeration. These approaches follow a Lagrangianperspective, in reference to fluid dynamics where the trajectory ofparticles is tracked over time. As such, they rely on accurate motionestimation, which is computationally expensive and difficult to makeartifact-free, especially at regions of occlusion boundaries andcomplicated motions. Moreover, the method of Liu et al. 2005 shows thatadditional techniques, including motion segmentation and imagein-painting, are required to produce good quality synthesis. Thisincreases the complexity of the algorithm further.

In contrast, the Eulerian perspective observes properties of a voxel offluid, such as pressure and velocity, which evolve over time. The methodstudies and amplifies the variation of pixel values over time, in aspatially-multiscale manner. The Eulerian approach to motionmagnification does not explicitly estimate motion, but ratherexaggerates motion by amplifying temporal color changes at fixedpositions. The method employs differential approximations that form thebasis of optical flow algorithms.

Temporal processing has been used previously to extract invisiblesignals, such as in Poh et al. 2010 (hereinafter “Poh”) and to smoothmotions, such as in Fuchs et al. 2010 (hereinafter “Fuchs”). Forexample, Poh describes extracting a heart rate from a video of a facebased on the temporal variation of the skin color, which is normallyinvisible to the human eye. Poh focuses on extracting a single number,whereas the method described herein employs localized spatial poolingand bandpass filtering to extract and reveal visually the signalcorresponding to the pulse. This primal domain analysis allowsamplification and visualization of the pulse signal at each location onthe face. This has important potential monitoring and diagnosticapplications to medicine, where, for example, the asymmetry in facialblood flow can be a symptom of arterial problems.

Fuchs uses per-pixel temporal filters to dampen temporal aliasing ofmotion in videos. Fuchs also describes the high-pass filtering ofmotion, but mostly for non-photorealistic effects and for large motions.In contrast, the method described herein strives to make imperceptiblemotions visible using a multiscale approach. The method described hereinexcels at amplifying small motions, in one embodiment.

First, nearly invisible changes in a dynamic environment can be revealedthrough Eulerian spatio-temporal processing of standard monocular videosequences. Moreover, for a range of amplification values that issuitable for various applications, explicit motion estimation is notrequired to amplify motion in natural videos. The method is robust andruns in real time. Second, an analysis of the link between temporalfiltering and spatial motion is provided and shows that the method isbest suited to small displacements and lower spatial frequencies. Third,a single framework can amplify both spatial motion and purely temporalchanges (e.g., a heart pulse) and can be adjusted to amplify particulartemporal frequencies—a feature which is not supported by Lagrangianmethods. Fourth, we analytically and empirically compare Eulerian andLagrangian motion magnification approaches under different noisyconditions. To demonstrate our approach, we present several exampleswhere our method makes subtle variations in a scene visible.

Space-Time Video Processing

FIG. 2 is an illustration of a temporal processing module employing theEulerian video magnification framework. The spatial decomposition module204 of system first decomposes the input video 202 into differentspatial frequency bands 206, then applies the same temporal filter tothe spatial frequency bands 206. The outputted filtered spatial bands210 are then amplified by an amplification factor 212, added back to theoriginal signal by adders 214, and collapsed by a reconstruction module216 to generate the output video 218. The temporal filter 208 andamplification factors 212 can be tuned to support differentapplications. For example, the system can reveal unseen motions of aDigital SLR camera, caused by the flipping mirror during a photo burst.

The method as illustrated by FIG. 2 combines spatial and temporalprocessing to emphasize subtle temporal changes in a video. The methoddecomposes the video sequence into different spatial frequency bands.These bands might be magnified differently because (a) they mightexhibit different signal-to-noise ratios or (b) they might containspatial frequencies for which the linear approximation used in motionmagnification does not hold. In the latter case, the method reduces theamplification for these bands to suppress artifacts. When the goal ofspatial processing is to increase temporal signal-to-noise ratio bypooling multiple pixels, the method spatially low-pass filters theframes of the video and downsamples them for computational efficiency.In the general case, however, the method computes a full Laplacianpyramid.

The method then performs temporal processing on each spatial band. Themethod considers the time series corresponding to the value of a pixelin a frequency band and applies a bandpass filter to extract thefrequency bands of interest. As one example, the method may selectfrequencies within the range of 0.4-4 Hz, corresponding to 24-240 beatsper minute, if the user wants to magnify a pulse. If the method extractsthe pulse rate, it can employ a narrow frequency band around that value.The temporal processing is uniform for all spatial levels and for allpixels within each level. The method then multiplies the extractedbandpassed signal by a magnification factor α. This factor can bespecified by the user, and may be attenuated automatically according toguidelines described later in this application. Next, the method addsthe magnified signal to the original signal and collapses the spatialpyramid to obtain the final output. Since natural videos are spatiallyand temporally smooth, and since the filtering is performed uniformlyover the pixels, the method implicitly maintains spatiotemporalcoherency of the results.

3 Eulerian Motion Magnification

The present method can amplify small motion without tracking motion asin Lagrangian methods. Temporal processing produces motionmagnification, shown using an analysis that relies on the first-orderTaylor series expansions common in optical flow analyses.

3.1 First-Order Motion

To explain the relationship between temporal processing and motionmagnification, the case of a 1D signal undergoing translational motionis informative. This analysis generalizes to locally-translationalmotion in 2D.

Function I(x, t) denotes the image intensity at position x and time t.Since the image undergoes translational motion, can be expressed as theobserved intensities with respect to a displacement function δ(t), I(x,t)=ƒ(x+δ(t)) and such that I(x, 0)=ƒ(x). The goal of motionmagnification is to synthesize the signal

Î(x,t)=ƒ(x+(1+α)δ(t))  (1)

for some amplification factor α. Assuming the image can be approximatedby a first-order Taylor series expansion, the image at time t, ƒ(x+δ(t))in a first-order Taylor expansion about x, can be written as

$\begin{matrix}{{I\left( {x,t} \right)} \approx {{f(x)} + {{\delta (t)}{\frac{\partial{f(x)}}{\partial x}.}}}} & (2)\end{matrix}$

Function B(x, t) is the result of applying a broadband temporal bandpassfilter to I(x, t) at every position x (picking out everything exceptƒ(x) in Eq. 2). For now, assume the motion signal, δ(t), is within thepassband of the temporal bandpass filter. Then,

$\begin{matrix}{\mspace{79mu} {{B\left( {x,\text{?}} \right)} = {{\delta (t)}{\frac{\partial{f(x)}}{\partial x}.\text{?}}\text{indicates text missing or illegible when filed}}}} & (3)\end{matrix}$

The method then amplifies that bandpass signal by α and adds it back toI(x, t), resulting in the processed signal

Ĩ(x,t)=I(x,t)+αB(x,t)  (4)

Combining Eqs. 2, 3, and 4 results in

$\begin{matrix}{{{I\left( {x,t} \right)} \approx {{f(x)} + {\left( {1 + \alpha} \right){\delta (t)}\frac{\partial{f(x)}}{\partial x}}}},} & (5)\end{matrix}$

Assuming the first-order Taylor expansion holds for the amplified largerperturbation, (1+α)δ(t), the amplification of the temporally bandpassedsignal can be related to motion magnification. The processed output issimply

Ĩ(x,t)=ƒ(x+(1+α)δ(t)).  (6)

This shows that the processing magnifies motions—the spatialdisplacement δ(t) of the local image ƒ(x) at time t has been amplifiedto a magnitude of (1+α).

FIG. 3 is a chart 300 illustrating that temporal filtering canapproximate spatial translation. This effect is demonstrated here on a1D signal, but applies to 2D. The input signal is shown at two timeinstants: I(x, t)=ƒ(x) at time t and I(x, t+1)=ƒ(x+δ) at time t+1. Thefirst order Taylor series expansion of I(x, t+1) about x approximatesthe translated signal. The temporal bandpass is amplified and added tothe original signal to generate a larger translation. In this exampleα=1, magnifying the motion by 100%, and the temporal filter is a finitedifference filter, subtracting the two curves.

This process is illustrated for a single sinusoid in FIG. 3. For a lowfrequency cosine wave and a relatively small displacement, δ(t), thefirst-order Taylor series expansion approximates the translated signalat time t+1. When boosting the temporal signal by α and adding it backto I(x, t), the Taylor series approximates that the expansion wave istranslated by (1+α)δ.

For completeness, in a more general case, δ(t) is not entirely withinthe passband of the temporal filter, δ_(k)(t), indexed by k, andrepresents the different temporal spectral components of δ(t). Eachδ_(k)(t) is attenuated by the temporal filtering by a factor λ_(k). Thisresults in a bandpassed signal,

$\begin{matrix}{\mspace{79mu} {{{B\left( {x,t} \right)} = {\text{?}(t)\frac{\partial{f(x)}}{\partial x}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (7)\end{matrix}$

(compare with Eq. 3). Because of the multiplication in Eq. 4, thistemporal frequency dependent attenuation can equivalently be interpretedas a frequency-dependent motion magnification factor, α_(k)=λ_(k)α.

3.2 Bounds

In practice, the assumptions in Sect. 3.1 hold for smooth images andsmall motions. For quickly changing image functions (i.e., high spatialfrequencies), ƒ(x), the first-order Taylor series approximations becomesinaccurate for large values of the perturbation, 1+αδ(t), whichincreases both with larger magnification α and motion δ(t). FIGS. 4 and5 demonstrate the effect of higher frequencies, larger amplificationfactors and larger motions on the motion-amplified signal of a sinusoid.

As a function of spatial frequency, ω, the process derives a guide forhow large the motion amplification factor, α, can be given the observedmotion δ(t). For the processed signal, Ĩ(x, t) to be approximately equalto the true magnified motion, Î(x, t), the process seeks the conditionsunder which

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} \approx {{\text{?}\left( {x,t} \right)\text{?}{f(x)}} + {\left( {1 + \alpha} \right){\delta (t)}\frac{\partial{f(x)}}{\partial x}}} \approx {f\left( {x + {\left( {1 + \alpha} \right){\delta (t)}}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}} & (8)\end{matrix}$

FIGS. 4A-D are charts illustrating motion amplification on a 1D signalfor different spatial frequencies and α values. For FIGS. 4A and 4C,λ=2π and

${\delta (1)} = \frac{\pi}{8}$

is the true translation. For FIGS. 4B and 4D, λ=π and

${\delta (1)} = {\frac{\pi}{8}.}$

FIGS. 4A-B illustrate the true displacement (e.g., true motionamplification, true motion application being represented byI(x,t)=f(x+(1+a)δ(t))) of I(x, 0) by (1+α)δ(t) at time t=1, colored fromblue (small amplification factor) to red (high amplification factor).FIG. 4C-D illustrate the amplified displacement (e.g., motionamplification via temporal filter, the motion amplification representedby I(x,t)=I(x,t)+αB(x,t)) produced by the filter of the present method,with colors corresponding to the correctly shifted signals in (a).Referencing Eq. 13, the red (far right) curves of each plot correspondto

${\left( {1 + \alpha} \right){\delta (t)}} = \frac{\lambda}{4}$

for the left plot, and

${\left( {1 + \alpha} \right){\delta (t)}} = \frac{\lambda}{2}$

for the right plot, showing the mild, then severe, artifacts introducedin the motion magnification from exceeding the bound on (1+α) by factorsof 2 and 4, respectively.

FIGS. 5A-B are charts 500 and 520, respectively, illustrating motionmagnification error, computed as the L₁-norm between the truemotion-amplified signal (e.g., FIGS. 4A-B) and the temporally-filteredresult (e.g., FIG. 4C-D), as function of wavelength, for variable valuesof δ(t) and a constant value of α, (FIG. 5A), and variable values of αand a constant value of δ(t), (FIG. 5B). In FIG. 5A, the constant valueof α is 1, and in FIG. 5B, the constant value of δ(t) is 2. The markerson each curve represent a derived cutoff point

$\begin{matrix}{{\left( {1 + \alpha} \right){\delta (t)}} = \frac{\lambda}{8}} & \left( {{Eq}.\mspace{14mu} 13} \right)\end{matrix}$

Let ƒ(x)=cos(ωx) for spatial frequency ω, and denote β=1+α. The processrequires that

cos(ωx)−βωδ(t)sin(ωx)≈cos(ωx+βωδ(t))  (9)

Using the addition law for cosines,

cos(ωx)−βωδ(t)sin(ωx)=cos(ωx)cos(βωδ(t))−sin(ωx)sin(βωδ(t))  (10)

Hence, the following should approximately hold

cos(βωδ(t))≈1  (11)

sin(βωδ(t))≈βδ(t)ω  (12)

FIG. 6 is a graph 600 illustrating a preferred amplification factor, α,as a function of spatial wavelength λ, for amplifying motion. Thepreferred amplification factor is fixed to a for spatial bands that arewithin the derived bound (Eq. 13), and is attenuated linearly for higherspatial frequencies.

The small angle approximations of Eqs. (11) and (12) hold to within 10%for

${{\beta\omega\delta}(t)} \leq \frac{\pi}{4}$

(the sine term is the leading approximation which gives

$\left. {{\sin \left( \frac{\pi}{4} \right)} = {0.9*\frac{\pi}{4}}} \right).$

In terms of the spatial wavelength,

${\lambda = \frac{2\pi}{\omega}},$

of the moving signal, this gives

$\begin{matrix}{{\left( {1 + \alpha} \right){\delta (t)}} < {\frac{\lambda}{8}.}} & (13)\end{matrix}$

Eq. 13 above provides the sought after guideline, giving the largestmotion amplification factor, α, compatible with accurate motionmagnification of a given video motion δ(t) and image structure spatialwavelength, λ. FIG. 4B shows the motion magnification errors for asinusoid when α is boosted beyond the limit in Eq. 13. In some videos,violating the approximation limit can be perceptually preferred and theλ cutoff is a user-modifiable parameter in the multiscale processing.

This suggests a scale-varying process: use a specified a magnificationfactor over some desired band of spatial frequencies, then scale backfor the high spatial frequencies (found from Eq. 13 or specified by theuser) where amplification gives undesirable artifacts. FIG. 6 shows sucha modulation scheme for a. Although areas of high spatial frequencies(sharp edges) are generally amplified less than lower frequencies, theresulting videos to contain perceptually appealing magnified motion.Such effect is also exploited in the earlier work of Freeman et al.[1991] to create the illusion of motion from still images.

Testing results can be generated using non-optimized MATLAB code on amachine with a six-core processor and 32 GB RAM. The computation timeper video is on the order of a few minutes. A separable binomial filterof size five constructs the video pyramids. A prototype applicationallows users to reveal subtle changes in real-time from live videofeeds, essentially serving as a microscope for temporal variations. Itis implemented in C++, can be CPU-based, and processes 640×480 videos at45 frames per second on a standard laptop, but can be furtheraccelerated by utilizing GPUs.

To process an input video by Eulerian video magnification, a user takesfour steps: (1) select a temporal bandpass filter; (2) select anamplification factor, α; (3) select a spatial frequency cutoff(specified by spatial wavelength, λ_(c)) beyond which an attenuatedversion of α is used; and (4) select the form of the attenuation forα—either force α to zero for all λ<λ_(c), or linearly scale a down tozero. The frequency band of interest can be chosen automatically in somecases, but it is often important for users to be able to control thefrequency band corresponding to their application. In our real-timeapplication, the amplification factor and cutoff frequencies are allcustomizable by the user.

FIGS. 7A-B are illustrations 700 and 710, respectively, showing Eulerianvideo magnification used to amplify subtle motions of blood vessels(e.g., a radial artery 714 and an ulnar artery 716) arising from bloodflow. For this video, the temporal filter is turned to a frequency bandthat includes the heart rate—0.88 Hz (53 bpm)—and the amplificationfactor is set to α=10. To reduce motion magnification of irrelevantobjects, a user-given mask amplifies the area near the wrist only.Movement of the radial artery 714 and the ulnar artery 716 can barely beseen in the input video of FIG. 7A taken with a standard point-and-shootcamera, but is significantly more noticeable in the motion-magnifiedoutput shown in FIG. 7B. The motion of the pulsing arteries is morevisible when observing a spatio-temporal YT slice 712 of the wrist (a)and (b).

FIG. 8 illustrates representative frames 810, 820, 830, 840, 850, and860 from additional videos demonstrating the process.

The process first selects the temporal bandpass filter to pull out themotions or signals to be amplified (step 1 above). The choice of filteris generally application dependent. For motion magnification, a filterwith a broad passband is preferred; for color amplification of bloodflow, a narrow passband produces a more noise-free result.

FIGS. 9A-D are graphs 900, 910, 920, and 930, respectively, illustratingthe frequency responses of a respective temporal filters. Ideal bandpassfilters are used for color amplification, since they have passbands withsharp cutoff frequencies. Low-order IIR filters can be useful for bothcolor amplification and motion magnification and are convenient for areal-time implementation. In general, two first-order lowpass IIRfilters with cutoff frequencies ω_(l) and ω_(h) are used to construct anIIR bandpass filter.

The process selects the desired magnification value, α, and spatialfrequency cutoff, λ_(c) (steps 2 and 3). While Eq. 13 can be a guide, inpractice, a user can try various α and λ_(c) values to achieve a desiredresult. The user can select a higher a that violates the bound toexaggerate specific motions or color changes at the cost of increasingnoise or introducing more artifacts. In some cases, the user can accountfor color clipping artifacts by attenuating the chrominance componentsof each frame. The method achieves this by doing all the processing inthe YIQ space. The user can attenuate the chrominance components, I andQ, before conversion to the original color space.

For human pulse color amplification, where emphasizing low spatialfrequency changes is essential, the process may force α=0 for spatialwavelengths below λ_(c). For motion magnification videos, the user canchoose to use a linear ramp transition for α (step 4).

The ideal filters, the frequency response of which is illustrated bygraphs 900 and 910 of FIGS. 9A-B, are implemented using a discretecosine transform (DCT). The graph 900 of FIG. 9A illustrates thefrequency response of an ideal filter from 0.8-1 Hz. The graph 900 is atransform of the “face” video, the representative frame 820 of which isillustrated in FIG. 8. The graph 910 of FIG. 9B illustrates thefrequency response of an ideal filter from 175-225 Hz. The graph 910 isa transform of the “guitar” video, the representative frame 830 of whichis illustrated in FIG. 8. The Butterworth filter, the frequency responseof which is illustrated by graph 920 of FIG. 9C, converts auser-specified frequency band to a second-order IIR structure and isused in our real-time application. The graph 920 of FIG. 9C illustratesthe frequency response of the Butterworth filter from 3.6-6.2 Hz. Thegraph 920 is a transform of the “subway” video, the representative frame840 of which is illustrated in FIG. 8. The second-order IIR filter, thefrequency response of which is illustrated by graph 930 of FIG. 9D, alsoallows user input to perform its transform. Second-order filters have abroader passband than an ideal filter. The graph 930 of FIG. 9Dillustrates the frequency response to a pulse detection.

The method was evaluated for color amplification using a few videos: twovideos of adults with different skin colors and one of a newborn baby.An adult subject with lighter complexion is shown in face (e.g., FIG.1), while an individual with darker complexion is shown in face2 (e.g.,FIG. 8). In both videos, the objective is to amplify the color change asthe blood flows through the face. In both face and face2, a Laplacianpyramid is applied and α is set for the finest two levels to 0.Essentially, each frame is downsampled and applied a spatial lowpassfilter to reduce both quantization and noise and to boost the subtlepulse signal. For each video, the method then passes each sequence offrames through an ideal bandpass filter with a passband of 0.83 Hz to 1Hz (50 bpm to 60 bpm). Finally, a large value of α≈100 and λ_(c)≈1000 isapplied to the resulting spatially lowpass signal to emphasize the colorchange as much as possible. The final video is formed by adding thissignal back to the original. The final video shows periodic green to redvariations at the heart rate and how blood perfuses the face.

baby2 is a video of a newborn recorded in situ at the Nursery Departmentat Winchester Hospital in Massachusetts. In addition to the video,ground truth vital signs from a hospital grade monitor are obtained.This information confirms the accuracy of the heart rate estimate andverifies that the color amplification signal extracted from our methodmatches the photoplethysmogram, an optically obtained measurement of theperfusion of blood to the skin, as measured by the monitor.

FIGS. 10A-B are diagrams 1000, and 1010 and 1020, respectively, ansynthetic input sequence (FIG. 10A) and respective selective motionamplification responses (FIG. 10B, 1010, 1020). The video sequencecontains blobs 1002, 1004, 1006, 1008 oscillating at temporalfrequencies of 7 Hz, 5 Hz, 3 Hz, and 2 Hz, respectively, as shown on theinput frame. The method uses an ideal temporal bandpass filter of 1-3 Hzto amplify only the motions occurring within the specified passband. InFIG. 10B, the spatio-temporal slices from the resulting video show thedifferent temporal frequencies and the amplified motion of the bloboscillating at 2 Hz. The space-time processing is applied uniformly toall the pixels.

To evaluate the method for motion magnification, several differentvideos are tested: face (FIG. 1), sim4 (FIG. 10), wrist (FIGS. 7A-B),camera (FIG. 2), face2, guitar, baby, subway, shadow, and baby2 (FIG.8). For all videos, a standard Laplacian pyramid for spatial filteringis used. For videos where motions at specific temporal frequencies(e.g., in sim4 and guitar) are to be emphasized, the method employsideal bandpass filters. In sim4 and guitar, the method can selectivelyamplify the motion of a specific blob or guitar string by using abandpass filter tuned to the oscillation frequency of the object ofinterest. These effects can be observed in the supplemental video. Thevalues used for α and λ_(c) for all of the videos are shown in Table 1.

For videos where revealing broad, but subtle motion is desired, themethod employs temporal filters with a broader passband. For example,for the face2 video, the method employs a second-order IIR filter withslow roll-off regions. By changing the temporal filter, the methodmagnifies the motion of the head rather than amplifying the change inthe skin color. Accordingly, α=20; λ_(c)=80 are chosen to magnify themotion.

By using broadband temporal filters and setting α and λ_(c) according toEq. 14, the method is able to reveal subtle motions, as in the cameraand wrist videos. For the camera video, a camera with a sampling rate of300 Hz records a Digital SLR camera vibrating while capturing photos atabout one exposure per second. The vibration caused by the moving mirrorin the SLR, though invisible to the naked eye, is revealed by ourapproach. To verify that the method amplifies the vibrations caused bythe flipping mirror, a laser pointer is secured to the camera and avideo of the laser light recorded, appearing at a distance of about fourmeters from the source. At that distance, the laser light visiblyoscillated with each exposure, with the oscillations in sync with themagnified motions.

The method exaggerates visible yet subtle motion, as seen in the baby,face2, and subway videos. In the subway example the method amplified themotion beyond the derived bounds of where the first-order approximationholds to increase the effect and to demonstrate the algorithm'sartifacts. Many testing examples contain oscillatory movements becausesuch motion generally has longer duration and smaller amplitudes.However, the method can amplify non-periodic motions as well, as long asthey are within the passband of the temporal bandpass filter. In shadow,for example, the method processes a video of the sun's shadow movinglinearly, yet imperceptibly, to the human eye over 15 seconds. Themagnified version makes it possible to see the change even within thisshort time period.

TABLE 1 Table 1: Table of α, λ_(c), ω_(t); ω_(h) values used to producethe various output videos. For face2, two different sets of parametersare used-one for amplifying pulse, another for amplifying motion. Forguitar, different cutoff frequencies and values for (α, λ_(c)) are usedto “select” the different oscillating guitar strings. f_(s) is the framerate of the camera. ω_(t) ω_(h) f_(s) Video α λ_(c) (Hz) (Hz) (Hz) baby10 16 0.4 3 30 baby2 150 600 2.33 2.67 30 camera 120 20 45 100 300 face100 1000 0.83 1 30 face2 20 80 0.83 1 30 motion face2 120 960 8.30 1 30pulse guitar 50 40 72 92 600 Low E guitar A 100 40 100 120 600 shadow 548 0.5 10 30 subway 60 90 3.6 6.2 30 wrist 10 80 0.4 3 30

Finally, videos may contain regions of temporal signals that do not needamplification, or that, when amplified, are perceptually unappealing.Due to Eulerian processing, the user can manually restrict magnificationto particular areas by marking them on the video.

Sensitivity to Noise. The amplitude variation of the signal of interestis often much smaller than the noise inherent in the video. In suchcases direct enhancement of the pixel values do not reveal the desiredsignal. Spatial filtering can enhance these subtle signals. However, ifthe spatial filter applied is not large enough, the signal of interestis not be revealed, as in FIG. 11.

Assuming that the noise is zero-mean white and wide-sense stationarywith respect to space, spatial low pass filtering reduces the varianceof the noise according to the area of the low pass filter. To boost thepower of a specific signal (e.g., the pulse signal in the face) thespatial characteristics of the signal can estimate the spatial filtersize.

Let the noise power level be σ², and the prior signal power level overspatial frequencies be S(λ). An ideal spatial low pass filter has radiusr such that the signal power is greater than the noise in the filteredfrequency region. The wavelength cut off of such a filter isproportional to its radius, r, so the signal prior can be represented asS(r). The noise power σ² can be estimated by examining pixel values in astable region of the scene, from a gray card. Since the filtered noisepower level, σ′², is inversely proportional to r², the radius r can befound with,

$\begin{matrix}{{S(r)} = {\sigma^{\prime 2} = {k\frac{\sigma^{2}}{r^{2}}}}} & (14)\end{matrix}$

where k is a constant that depends on the shape of the low pass filter.This equation gives an estimate for the size of the spatial filterneeded to reveal the signal at a certain noise power level.

Eulerian vs. Lagrangian Processing. Because the two methods takedifferent approaches to motion—Lagrangian approaches explicitly trackmotions, while Eulerian approach does not—they can be used forcomplementary motion domains. Lagrangian approaches enhance motions offine point features and support larger amplification factors, while theEulerian method is better suited to smoother structures and smallamplifications. The Eulerian method does not assume particular types ofmotions. The first-order Taylor series analysis can hold for generalsmall 2D motions along general paths.

FIG. 11A illustrates frame from the face video (FIG. 1) with whiteGaussian noise (σ=0.1 pixel) added. FIG. 11B illustrates intensitytraces over time for the pixel marked blue on the input frame, showingthe trace obtained when the (noisy) sequence is processed with the samespatial filter used to process the original face sequence, a separablebinomial filter of size 20. FIG. 11C shows the trace when using a filtertuned according to the estimated radius in Eq. 14, a binomial filter ofsize 80. The pulse signal is not visible in FIG. 11B, as the noise levelis higher than the power of the signal, while in FIG. 11C the pulse isclearly visible (the periodic peaks about one second apart in thetrace).

Appendix A provides estimates of the accuracy of the two approaches withrespect to noise. Comparing the Lagrangian error, εL (Eq. 28), and theEulerian error, εE (Eq. 30), both methods are equally sensitive to thetemporal characteristics of the noise, n_(t), while the Lagrangianprocess has additional error terms proportional to the spatialcharacteristics of the noise, n_(x), due to the explicit estimation ofmotion (Eq. 26). The Eulerian error, on the other hand, growsquadratically with α, and is more sensitive to high spatial frequencies(I_(xx)). In general, this means that Eulerian magnification ispreferable over Lagrangian magnification for small amplifications andlarger noise levels.

This analysis is validated on a synthetic sequence of a 2D cosineoscillating at 2 Hz temporally and 0.1 pixels spatially with additivewhite spatiotemporal Gaussian noise of zero mean and standard deviationσ (FIG. 12). The results match the error-to-noise anderror-to-amplification relationships predicted by the derivation (FIG.12B)). The region where the Eulerian approach outperforms the Lagrangianresults (FIG. 12A, left) is also as expected. The Lagrangian method ismore sensitive to increases in spatial noise, while the Eulerian erroris hardly affected by it (FIG. 12C). While different regularizationschemes used for motion estimation (that are harder to analyzetheoretically) may alleviate the Lagrangian error, they do not changethe result significantly (FIG. 12A, right). In general, for smallamplifications, the Eulerian approach strikes a better balance betweenperformance and efficiency.

The method takes a video as input and exaggerates subtle color changesand imperceptible motions. To amplify motion, the method does notperform feature tracking or optical flow computation, but merelymagnifies temporal color changes using spatio-temporal processing. ThisEulerian based method, which temporally processes pixels in a fixedspatial region, successfully reveals informative signals and amplifiessmall motions in real-world videos.

Eulerian and Lagrangian Error

This section estimates the error in the Eulerian and Lagrangian motionmagnification with respect to spatial and temporal noise. The derivationis performed again in 1D for simplicity, and can be generalized to 2D.

Both methods approximate the true motion-amplified sequence, Î(x, t), asshown in (1). First, analyze the error in those approximations on theclean signal, I(x, t).

Without noise: In the Lagrangian approach, the motion amplifiedsequence, Ĩ_(L)(x, t), is achieved by directly amplifying the estimatedmotion, (t), with respect to the reference frame, I(x, 0)

Ĩ _(L)(x,t)=I(x+(1+α){tilde over (δ)}(t),0)  (15)

In its simplest form, δ(t) can be estimated in a point-wise manner

$\begin{matrix}{{{\text{?}(t)} = \text{?}}{\text{?}\text{indicates text missing or illegible when filed}}} & (16)\end{matrix}$

where I_(x)(x, t)=∂I(x, t)/∂x and I_(t)(x, t)=I(x, t)−I(x, 0). From nowon, the space (x) and time (t) indices are omitted when possible forbrevity.

The error in the Lagrangian solution is directly determined by the errorin the estimated motion, which is the second-order term in thebrightness constancy equation,

$\begin{matrix}{\mspace{79mu} {{{I\left( {x,t} \right)} \approx {{I\left( {x,0} \right)} + {{\delta (t)}I_{x}} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}} \approx {{\delta \text{?}} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (17)\end{matrix}$

The estimated motion, {tilde over (δ)}(t), is related to the truemotion, δ(t), by

$\begin{matrix}{\mspace{79mu} {{{\overset{\sim}{\delta}(t)} \approx {{\text{?}(t)} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (18)\end{matrix}$

Plugging (18) in (15) and using a Taylor expansion of/about x+(1+α)δ(t)gives,

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} \approx {{I\left( {{x + {\left( {1 + \alpha} \right)\text{?}(t)}},0} \right)} + {\frac{1}{2}\left( {1 + \alpha} \right)\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (19)\end{matrix}$

Subtracting (1) from (19), the error in the Lagrangian motion magnifiedsequence, εL, is

$\begin{matrix}{\text{?} \approx {{{{\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}}}.\text{?}}\text{indicates text missing or illegible when filed}}} & (20)\end{matrix}$

In the Eulerian approach, the magnified sequence, Î_(E)(x, t), is

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} = {{{I\left( {x,t} \right)} + {\alpha \; \text{?}\left( {x,t} \right)}} = {{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\text{?}\left( {x,t} \right)}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (21)\end{matrix}$

similar to (4), using a two-tap temporal filter to compute I_(t). Usinga Taylor expansion of the true motion-magnified sequence, Î defined in(1), about x gives,

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} \approx {{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right){\delta (t)}\text{?}} + {\frac{1}{2}\left( {1 + \alpha} \right)^{2}\delta^{2}\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (22)\end{matrix}$

Using (17) and subtracting (1) from (22), the error in the Eulerianmotion-magnified sequence, εE, is

$\begin{matrix}{\text{?} \approx {{{\frac{1}{2}\left( {1 + \alpha} \right)^{2}{\delta^{2}(t)}\text{?}} - {\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}\text{?}\text{indicates text missing or illegible when filed}}}}} & (23)\end{matrix}$

With noise: Let I′(x, t) be the noisy signal, such that

I′(x,t)=I(x,t)+n(x,t)  (24)

for additive noise n(x; t).

The estimated motion in the Lagrangian approach becomes

$\begin{matrix}{{{\text{?}(t)} = {\text{?} = \text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (25)\end{matrix}$

where n_(x)=∂_(n)/∂_(x) and n_(t)=n(x, t)−n(x, 0). Using a TaylorExpansion on (n_(t), n_(x)) about (0, 0) (zero noise), and using (17),gives

$\begin{matrix}{{{\text{?}(t)} \approx {{\delta (t)} + \text{?} - \text{?} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (26)\end{matrix}$

Plugging (26) into (15), and using a Taylor expansion of I aboutx+(1+α)δ(t), gives

$\begin{matrix}{\text{?}\left( {1 + \alpha} \right)\text{?}\left( {\text{?} - \text{?} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}} \right){n.\text{?}}\text{indicates text missing or illegible when filed}} & (27)\end{matrix}$

Using (18) again and subtracting (1), the Lagrangian error as a functionof noise, εL(n), is

$\begin{matrix}{\text{?}{{{{\left( {1 + \alpha} \right)\text{?}} - {\left( {1 + \alpha} \right)\text{?}{\delta (t)}} - {\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}} + {\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}} + n}}.\text{?}}\text{indicates text missing or illegible when filed}} & (28)\end{matrix}$

In the Eulerian approach, the noisy motion-magnified sequence becomes

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} = {{{\text{?}\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\text{?}}} = {{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\left( {\text{?} + \text{?}} \right)} + n}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (29)\end{matrix}$

Using (23) and subtracting (1), the Eulerian error as a function ofnoise, εE(n), is

$\begin{matrix}{\text{?}{\text{?}\text{indicates text missing or illegible when filed}}} & (30)\end{matrix}$

If the noise is set to zero in (28) and (30), the resulting errorscorrespond to those derived for the non-noisy signal as shown in (20)and (23).

FIGS. 12A-C are diagrams illustrating a comparison between Eulerian andLagrangian motion magnification on a synthetic sequence with additivenoise. FIG. 12A shows diagrams 1202 and 1204 illustrating the minimalerror, min(εE, εL), computed as the (frame-wise) RMSE between eachmethod's result and the true motion-magnified sequence, as function ofnoise and amplification, colored from blue (small error) to red (largeerror). Graph 1202 illustrates the minimal error with spatialregularization in the Lagrangian method, and graph 1204 illustrates theminimal error without spatial regularization in the Lagrangian method.The black curves mark the intersection between the error surfaces, andthe overlayed text indicate the best performing method in each region.FIG. 11B shows RMSE of the two approaches as function of noise (left)and amplification (right).

In this section Supplemental Information estimates of the error in theEulerian and Lagrangian motion magnification results with respect tospatial and temporal noise are derived. The derivation is done again forthe 1D case for simplicity, and can be generalized to 2D. The truemotion-magnified sequence is

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} = {{\text{?}\left( {x + {\left( {1 + \alpha} \right){\delta (t)}}} \right)} = {I\left( {{x + {\left( {1 + \alpha} \right){\delta (t)}}},0} \right)}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (31)\end{matrix}$

Both methods approximate the true motion-amplified sequence, Î(x, t)(Eq. 31). First, the error in those approximations on the clean signal,I(x, t), is analyzed.

1.1 Without Noise

In the Lagrangian approach, the motion-amplified sequence, Ĩ_(L)(x, t),is achieved by directly amplifying the estimated motion, {tilde over(δ)}(t), with respect to the reference frame I(x, 0)

Ĩ _(L)(x,t)=I(x+(1+α){tilde over (δ)}(t),0)  (32)

In its simplest form, δ(t) can be estimated using point-wise brightnessconstancy

$\begin{matrix}{{{\text{?}(t)} = \text{?}}{\text{?}\text{indicates text missing or illegible when filed}}} & (33)\end{matrix}$

where I_(x)(x, t)=∂I(x, t)/∂x and I_(t)(x, t)=I(x, t)−I(x, 0). From nowon, space (x) and time (t) indices are omitted when possible forbrevity.

The error in the Lagrangian solution is directly determined by the errorin the estimated motion, which is the second-order term in thebrightness constancy equation

$\begin{matrix}{{{I\left( {x,t} \right)} = {{I\left( {{x + {\delta (t)}},0} \right)} \approx {{I\left( {x,0} \right)} + {{\delta (t)}\text{?}} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}} \approx {{\delta (t)} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (34)\end{matrix}$

So that the estimated motion {tilde over (δ)}(t) is related to the truemotion, δ(t), as

$\begin{matrix}{{\text{?} + {\frac{1}{2}\text{?}(t)\text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (35)\end{matrix}$

Plugging (35) in (32),

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} \approx {I\left( {{x + {\left( {1 + \alpha} \right)\left( {{\delta (t)} + {\frac{1}{2}\text{?}(t)\text{?}}} \right)}},0} \right)} \approx {I\left( {x + {\left( {1 + \alpha} \right){\delta (t)}} + {\frac{1}{2}\left( {1 + \alpha} \right)\text{?}(t)\text{?}0}} \right)}}{\text{?}\text{indicates text missing or illegible when filed}}} & (36)\end{matrix}$

Using first-order Taylor expansion of I about x+(1+α)δ(t),

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} \approx {{I\left( {{x + {\left( {1 + \alpha} \right){\delta (t)}}},0} \right)} + {\frac{1}{2}\left( {1 + \alpha} \right)\text{?}(t)\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (37)\end{matrix}$

Subtracting (31) from (37), the error in the Lagrangian motion-magnifiedsequence, εL, is

$\begin{matrix}{{\text{?} \approx {{\frac{1}{2}\left( {1 + \alpha} \right)\text{?}(t)\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (38)\end{matrix}$

In the Eulerian approach, the magnified sequence, Î_(E)(x, t), iscomputed as

$\begin{matrix}{\mspace{79mu} {{{{\hat{I}}_{E}\left( {x,t} \right)} = {{{I\left( {x,t} \right)} + {\alpha \text{?}\left( {x,t} \right)}} = {{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\text{?}\left( {x,t} \right)}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (39)\end{matrix}$

similar to Eq. 34, using a two-tap temporal filter to compute I_(t).

Using Taylor expansion of the true motion-magnified sequence, Î (Eq. 1),about x, gives

$\begin{matrix}{\mspace{79mu} {{{I\left( {x,t} \right)} \approx {{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right){\delta (t)}I_{x}} + {\frac{1}{2}\left( {1 + \alpha} \right)^{2}\text{?}(t)\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (40)\end{matrix}$

Plugging (34) into (40)

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} \approx {{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\left( {\text{?} - {\frac{1}{2}\text{?}(t)\text{?}}} \right)} + {\frac{1}{2}\left( {1 + \alpha} \right)^{2}\text{?}(t)\text{?}}} \approx {{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\text{?}} - {\frac{1}{2}\left( {1 + \alpha} \right)\text{?}(t)\text{?}} + {\frac{1}{2}\left( {1 + \alpha} \right)^{2}\text{?}(t)\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (41)\end{matrix}$

Subtracting (39) from (41) gives the error in the Eulerian solution

$\begin{matrix}{{{\text{?}{}} \approx {{\frac{1}{2}\left( {1 + \alpha} \right)^{2}\text{?}(t)\text{?}} - {\frac{1}{2}\left( {1 + \alpha} \right)\text{?}(t)\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (42)\end{matrix}$

1.2 With Noise

Let I′(x, t) be the noisy signal, such that

I′(x,t)=I(x,t)+n(x,t)  (43)

for additive noise n(x, t).

In the Lagrangian approach, the estimated motion becomes

$\begin{matrix}{\mspace{79mu} {{{\hat{\delta}(t)} = {\text{?} = \text{?}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (44)\end{matrix}$

where n_(x)=∂n/∂x and n_(t)=n(x, t)−n(x, 0).

Using Taylor Expansion on (n_(t), n_(x)) about (0, 0) (zero noise), andusing (34) gives,

$\begin{matrix}{{{\text{?}(t)} \approx {\text{?} + {\text{?}\text{?}}} \approx {{\delta (t)} + \text{?} - \text{?} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (45)\end{matrix}$

where the terms involving products of the noise components are ignored.

Plugging into Eq. (32), and using Taylor expansion of I aboutx+(1+α)δ(t) (t) gives

$\begin{matrix}\left. {\left. {{{\overset{\sim}{I}}_{L}^{\prime}\left( {x,t} \right)} \approx {{I\left( {{x + {\left( {1 + \alpha} \right){\delta (t)}}},0} \right)} + {\left( {1 + \alpha} \right){I_{x}\left( {\frac{n_{t}}{I_{x}} - {n_{x}\frac{I_{t}}{I_{x}^{2}}} + {\frac{1}{2}I_{xx}{\delta^{2}(t)}}} \right)}}}} \right) + n} \right) & (46)\end{matrix}$

Arranging terms, and substituting (34) in (46),

${{{\overset{\sim}{I}}_{L}^{\prime}\left( {x,t} \right)} \approx {{I\left( {{x + {\left( {1 + \alpha} \right){\delta (t)}}},0} \right)} + {\left( {1 + \alpha} \right)\left( {\text{?} - {\text{?}\left( {{\text{?}(t)} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}} \right)} + {\frac{1}{2}{\delta^{2}(t)}\text{?}}} \right)} + n}} = {{I\left( {{x + {\left( {1 + \alpha} \right){\delta (t)}}},0} \right)} + {\left( {1 + \alpha} \right)\text{?}} - {\left( {1 + \alpha} \right)\text{?}(t)} - {\frac{1}{2}\left( {1 + \alpha} \right)\text{?}{\delta^{2}(t)}\text{?}} + {\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}} + {n\text{?}\text{indicates text missing or illegible when filed}}}$

(47)

Using (35) again and subtracting (31), the Lagrangian error as functionof noise, εL(n), is

${ɛ_{L}(n)} \approx {{{\left( {1 + \alpha} \right)\text{?}} - {\left( {1 + \alpha} \right)\text{?}{\delta (t)}} - {\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}} + {\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}} + n}}$?indicates text missing or illegible when filed

(48)

In the Eulerian approach, the noisy motion-magnified sequence becomes

$\begin{matrix}{{{\text{?}\left( {x,t} \right)} = {{{I^{\prime}\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\text{?}}} = {{{I\left( {x,0} \right)} + {\left( {1 + \alpha} \right)\left( {\text{?} + \text{?}} \right)} + n} = {{\text{?}\left( {x,t} \right)} + {\left( {1 + \alpha} \right)\text{?}} + n}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (49)\end{matrix}$

Using (42) and subtracting (31), the Eulerian error as function ofnoise, εE(n), is

$\begin{matrix}{{{\text{?}(n)} \approx {{{\left( {1 + \alpha} \right)\text{?}} + {\frac{1}{2}\left( {1 + \alpha} \right)^{2}{\delta^{2}(t)}\text{?}} - {\frac{1}{2}\left( {1 + \alpha} \right){\delta^{2}(t)}\text{?}} + n}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (50)\end{matrix}$

Notice that setting zero noise in (48) and (50), gives the correspondingerrors derived for the non-noisy signal in (38) and (42).

FIG. 13 is a flow diagram 1300 illustrating an example embodiment of theEulerian embodiment of the method described herein. The Eulerian-basedmethod begins by examining pixel values of two or more images (1302).The method then determines the temporal variation of the examined pixelvalues (1304). The method is designed to amplify only small temporalvariations. While the method can be applied to large temporalvariations, the advantage in the method is provided for small temporalvariations. Therefore the method is optimized when the input video hassmall temporal variations between the images. The method then appliessignal processing to the pixel values (1306). For example, signalprocessing can amplify the determined temporal variations, even when thetemporal variations are small.

FIG. 14 illustrates a computer network or similar digital processingenvironment in which the present invention may be implemented.

Client computer(s)/devices 50 and server computer(s) 60 provideprocessing, storage, and input/output devices executing applicationprograms and the like. Client computer(s)/devices 50 can also be linkedthrough communications network 70 to other computing devices, includingother client devices/processes 50 and server computer(s) 60.Communications network 70 can be part of a remote access network, aglobal network (e.g., the Internet), a worldwide collection ofcomputers, Local area or Wide area networks, and gateways that currentlyuse respective protocols (TCP/IP, Bluetooth, etc.) to communicate withone another. Other electronic device/computer network architectures aresuitable.

FIG. 15 is a diagram of the internal structure of a computer (e.g.,client processor/device 50 or server computers 60) in the computersystem of FIG. 14. Each computer 50, 60 contains system bus 79, where abus is a set of hardware lines used for data transfer among thecomponents of a computer or processing system. Bus 79 is essentially ashared conduit that connects different elements of a computer system(e.g., processor, disk storage, memory, input/output ports, networkports, etc.) that enables the transfer of information between theelements. Attached to system bus 79 is I/O device interface 82 forconnecting various input and output devices (e.g., keyboard, mouse,displays, printers, speakers, etc.) to the computer 50, 60. Networkinterface 86 allows the computer to connect to various other devicesattached to a network (e.g., network 70 of FIG. 14). Memory 90 providesvolatile storage for computer software instructions 92 and data 94 usedto implement an embodiment of the present invention (e.g., code detailedabove). Disk storage 95 provides non-volatile storage for computersoftware instructions 92 and data 94 used to implement an embodiment ofthe present invention. Central processor unit 84 is also attached tosystem bus 79 and provides for the execution of computer instructions.

In one embodiment, the processor routines 92 and data 94 are a computerprogram product (generally referenced 92), including a computer readablemedium (e.g., a removable storage medium such as one or more DVD-ROM's,CD-ROM's, diskettes, tapes, etc.) that provides at least a portion ofthe software instructions for the invention system. Computer programproduct 92 can be installed by any suitable software installationprocedure, as is well known in the art. In another embodiment, at leasta portion of the software instructions may also be downloaded over acable, communication and/or wireless connection. In other embodiments,the invention programs are a computer program propagated signal product107 embodied on a propagated signal on a propagation medium (e.g., aradio wave, an infrared wave, a laser wave, a sound wave, or anelectrical wave propagated over a global network such as the Internet,or other network(s)). Such carrier medium or signals provide at least aportion of the software instructions for the present inventionroutines/program 92.

In alternate embodiments, the propagated signal is an analog carrierwave or digital signal carried on the propagated medium. For example,the propagated signal may be a digitized signal propagated over a globalnetwork (e.g., the Internet), a telecommunications network, or othernetwork. In one embodiment, the propagated signal is a signal that istransmitted over the propagation medium over a period of time, such asthe instructions for a software application sent in packets over anetwork over a period of milliseconds, seconds, minutes, or longer. Inanother embodiment, the computer readable medium of computer programproduct 92 is a propagation medium that the computer system 50 mayreceive and read, such as by receiving the propagation medium andidentifying a propagated signal embodied in the propagation medium, asdescribed above for computer program propagated signal product.

Generally speaking, the term “carrier medium” or transient carrierencompasses the foregoing transient signals, propagated signals,propagated medium, storage medium and the like.

While this invention has been particularly shown and described withreferences to example embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims.

What is claimed is:
 1. A method of amplifying temporal variation in atleast two images, the method comprising: examining pixel values of theat least two images, the temporal variation of the pixel values betweenthe at least two images being below a particular threshold; applyingsignal processing to the pixel values.
 2. The method of claim 1, whereinapplying signal processing amplifies the variations of the pixel valuesbetween the at least two images.
 3. The method of claim 1, wherein thesignal processing is temporal processing.
 4. The method of claim 3,wherein the temporal processing is a bandpass filter.
 5. The method ofclaim 4, wherein the bandpass filter is configured to analyzefrequencies over time.
 6. The method of claim 1, wherein applying signalprocessing includes spatial processing.
 7. The method of claim 6,wherein the spatial processing removes noise.
 8. The method of claim 1,further comprising visualizing a pattern of flow of blood in a bodyshown in the at least two images.
 9. A system for amplifying temporalvariation in at least two images, the system comprising: a pixelexamination module configured to examine pixel values of the at leasttwo images, the temporal variation of the pixel values between the atleast two images being below a particular threshold; a signal processingmodule configured to apply signal processing to the pixel values. 10.The system of claim 9, wherein the signal processing module is furtherconfigured to amplify variations of the pixel values between the atleast two images.
 11. The system of claim 9, wherein the signalprocessing is temporal processing.
 12. The system of claim 11, whereinthe temporal processing is a bandpass filter.
 13. The system of claim12, wherein the bandpass filter is configured to analyze frequenciesover time.
 14. The system of claim 9, wherein applying signal processingincludes spatial processing.
 15. The system of claim 14, wherein thespatial processing removes noise.
 16. The system of claim 9, furthercomprising a blood flow visualization module configured to visualize apattern of flow of blood in a body shown in the at least two images.